Welp. That's done it.

I reflashed the router and everything came back the way it should have been from the beginning.

So, the lesson learned:

When repurposing consumer networking kit for tasks it isn't meant for (i.e., using a router for a bridge), remember to reflash it when you use it for its intended purpose.

Apparently, the Alchemy ROM from Sveasoft hides a very important option, or I completely overlooked it. Linksys has a very convenient setting called "AP Isolation" on the WRT54G. As you might imagine, it basically creates a VLAN for every wirelessly connected computer. These VLANs don't easily route to each other. During the reconfiguration process, I noticed that setting and made sure it was switched to "Off". The problem is no more.

I'm half tempted to reflash to the Alchemy ROM to see if that setting is an option or not, so that I can know if I'm stupid.

So, if anyone has experience with the Alchemy ROM go to your advanced wireless settings and see if there is something that would lead you to believe that I'm an idiot. My wager is not, since the Alchemy ROM has all of those advanced VLAN configuration screens (all of which I'm pretty sure were configured to make our network a single VLAN).

So, thanks a lot guys. You gave me a lot of places to look that I just wasn't catching, and lead me away from dead ends.

Peace,
Dhej